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Policy, Resewh, and ixtmal Affairs
WORKING PAPERS
Pubic Eoonomics
Country Economics DepartmentThe World Bank
April 1990WPS 410
The Cost of Capitaland Investment
in Developing Countries
Alan Auerbach
A model for evaluating how policy changes might affect incen-tives to invest in developing countries.
The Policy, Rearch, and Extemal Affairs Complex distributes PRE Wohkuig Papens to dcissatinate the findings of work in psogre6sand to encourage the exchange of ideas among Bank staff and all others interested in development issues. These papas carry the namesof the authors, reflect only their views, and should he used and cited accordingly The findings, interpretations, and conclusions are theauthors' own 1 hey should not he attnbuted to thc World Bank, its Board of Directors, its management, or any of its mcmber countines
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Polley, Roellh, and,Externa Affnlrn
Pubilc Economics|
This paper- a product of the Public Economics Division, Country Economics Department - is part ofa larger effon in PRE to promote sound public policies in the development of the private sector indeveloping countries. Copies are available free from the World Bank, 1818 H Street NW, Washington DC20433. Please contact Ann Bhalla, room NIO-059, extension 37699 (43 pages).
Auerbach's model can be used to evaluate how For each dollar of forgone tax revenues, howcurrent and new policies might affect incentives much have tax incentives stimulated investment?to invest in a developing country.
Do taxes af'ect foreign investment inIt takes into account factors that are often developing countries? Do they influence foreign
ignored in analyses of investment in more business locations? If they do, what instrumentsdeveloped countries - such factors as risk, best stimulate the most investment per dollar offoreign tax provisions that affect capital flows, tax revenues lost to the host country?the prevalence of trade distortions, the lack ofdomestic capital markets, and the relative How do taxes affect industrial production?credibility of government policy changes. How have tax instruinents affected the use of
labor? How have they affected physical, re-The author reviews the literature on invest- search, and development capital?
ment and the cost of capital, showing how theeffects of tax and nontax government policies How have business taxes and tax expendi-should be incorporated in any analysis of tures (forgone revenues) affected technologicalinvestment behavior. change, the expansion of private output, and
after-tax profits?The methodology is more general than
calculations of tax wedges and effective tax Are these tax-induced distortions preventingrates. It should help developing countries firms from holding optimal levels of fixedmeasure the efficacy of current policies, predict factors?how policy changes may influence investment,and determine appropriate directions for reform. Given the empirical estimates obtained in
this study on factor substitution, a bias towardThis paper is the first in a series of papers technical change, and scale economies, wlhat
commissioned by the Tax Incentives for Indus- revenue-neutral altemative tax policy environ-trial and Technological Development Research ment would encourage growth and productivity?Project of the Public Economics Division.Rescarchers on this project have focused on thefollowing questions:
The PRE Working Paper Scries disseminates thc findings of work under way in thc Bank's Policy, Research, and ExternalAffairs Complex. An objective of the series is to get these findings out quickly, even if presentations are less than fullypolished. The findings, interpretations, and conclusions in these papers do not necessarily reprcsent official Bank policy.
Produced at thc PRE Dissemination Center
THE COST OF CAPITAL AND IUVYSTINT IN DEVELOPING COUNTRIFS
by
Alan Auerbach
Table of Contents
1. Introduction ... 0 * .0..... ... ...........................1
II. The Model ............................................... 2
A. The User Cost of Capital............................ 2B. The Effective Tax Rato.............................. 6C. The Effective Tax Rate and the User Cost of Capital,
More Broadly Deffned ........................ ... ... 11De Summaryot ....... *se.. 99eoo e 16
III. Changes in Tax Regime.....o................e 17
A. Changes in Tax St cture .................. 21B. Tax Holidays ............... .............. 22C. Tax Law Asymetries ............ ............... . 23D. Uncertainty and Risk ............................... . 25
IV Institutional Factors: Calculat'ng r and r....e.......... 27
A. Measuring Investment Incentives........... ........... 27B. Mcasu:ing the Required Rate of Return ........ ....... 29
e. Conclusions ........................................... . . 33
Tables ..... .................. 34Appendix ................................................... 36References ......... ......................................... 39Footnotes ... ............................................. 41
I. introduction
For governments ln developLng countrles, an important policy objeetlve
is the creatlon of an environment that attracts capital to high-return fixed
investment projects. Like more advanced countries, these sconomies seek the
increases in labor productivity and living standArde that capital deepening
brings. For many reasons, however, the design of government policy toward
investment in less developed countries is both more critical and more complex.
First, such countries may lack fully functioning internal capital
markets, making it difficult to measure the cost of capital for new projects.
Second, the inadequacy of domestic capital may force a significant dependence
on foreign direct investment, which requires a more complete involvement on
the part of the investor than simply supplying funds. Third, such countrLes
typically impose more significant trade and production distortions in the form
of excise taxes, tariffs, quotas snd restrictions, for which account must be
taken in estimating the incentives facing a potential investment. Fourth,
certain types of investment incentives require an administrative
infrastructure that may be absent in all but the most developed countries.
Finally, the governments in these countries may lack the credibility needed to
convince investors to respond to an announced change in policy.
Though the problems of policy design are considerable, so are the
potential social returns from an appropriate investment climate. This paper
develops measures of the incentive to invest that can be used to evaluate
existing policies and guide the design of new ones, taking account of the
complications just mentioned. The next section introduces the basic notation
and modelling assumptions, while subsequent sections develop the model and its
implications more fully.
1
IL Tb. 3.1
A. Th Usar Coat of Canital
To evaluate the incentive to invest, we consider the decisions of a firm
that uses a single capital Input, labor and intermediate inputs in the
production process. The simplifying assumptions that capital and labor are
homogenevous are not critical for most purposes of analysis. Initially, we
vill also assume that the firm's investments are riLkless, and that it faces a
constant tax system with full loss offset, ha perfect certainty about the
future, may adjust capital instantaneously, and is perfectly competitive (i.e.
takes all prices as given). Though these restrictions are often made in
analyzing investment incentives, they influence the results considerably and
are particularly inappropriate in the present context. They are imposed
initially for purposes of exposition and to permit a comparison of this
paper's approach to those found elsewhere.
The model's notation is summarized in Table 1. We examine a firm that
produces gross output X using capital, K, labor, L and inputs M according to
the following relationship:
(1) X - X(K,L,M)
where X(.) is a general production function with nonincreasing returns to
scale.
Let r be the real discount rate that the firm uses in valuing future cash
flows from the investment project. As discussed in Auerbach (1983b), this may
be constructed as a weighted average of the real costs of debt and equity
finance. For example, in a closed economy without an indexed tax system, the
2
formula for v would be:
(2) r - bL(Il-.) - ws + (l-b)p/(1 )
vhere 1 is the nomlnal lnterest rate, r is the corporate tax rate against
which interest payments are normally tax deductible, is l the real discount
rate of equlty-holders, * ls the tax rate effectively applied to real equlty
returns, a is the inflation rate, and b is the fraction of the project
flnanced vith debt. The construction of thls measure depends on a nuober of
lnstitutional factors, such as the source of marglnal equity funds, for this
determines the extent to which the tax rate on dividends actually exerts a
marginal impact. In less developed countries, calculation of the relevant
interest rate, as well as the lmportance of foreign lnvestors, may be more
slgnlflcant questlons. These lssues are dlscussed further below. For the
moment, the analysls simply takes the de nrmination of r as given. Let the
prlces of output, materlals and capltal goods that the company faces be p, v
and g, respectlvely, and let w be the wage rate. Because of taxes and other
distortions ln product and Lactor markets, these wlll not necessarily be
observed *marketu prlces. They should simply be lnterpreted as the effective
marginal prlces that firms face for the assoclated commodities.
Let r be the present value of the after-tax cash flow attrlbutable to
depreciatLon allowances, lnvestment grants and lnvestment tax credits received
by the firm per dollar of new investment. That is, if grants and credits k
are recelved immediately and depreciatlon deductions D(t-s) are received at
each date t after the initial lnvestment at date s, then
rZ -(r+w)(t-s)(3) r - k + rz - k + Js f... D(t-s) dt
3
There are many types of investmnt incentives used in practice. Though some
ore more complicated, most can be expressed using this framework. This is
discussed below.
The zorporat!on's problem of maximizing the wealth of its shareholders at
date s may then be shown (Auerbach 198? e to be equivalent to maximizing:
a -(r+wr (t-s)(4) Vs Js a lr)lptX(KtLtMt)-tLt-vtMt gt(1-n)It)dt
where Itis the firm's investment at date t. Under the famIliar assumption
that capital decays exponentially at rate 6, the avolution of the capital
stock obeys the expression:
(5) Kt K It , 6Kt
The firm chooses I, L, and M at each date after t in order to maximize
the function V . In order to focus on the investment decision, it will
sometimes be useful to consider this decision conditional oo the optimal
decisions with respect to labor and material inputs. Since each is a vatiable
factor of production, the optimization produces for each yields the standard
rule of setting equal contemporaneous marginal revenues and costs at each date
t > s:
(6) XLt wpt
(7) x -t m vt/Pt
4
The decision rules (6) and (7) provide two equations in the variables L,
X sod K. Hence, they may be used to define L and K implicitly in terms of K.
That _J, froo& (5) and (7) we may obtain expressions:
(8) L* - L(K, w/p, v/p)
(9) M* - M(K, w/p, v/p)
which may be used to obtain a production function of K alone:
(10) F_(K) - X(K,L*,1H*) Lt
A time subscript must be attached to the new function F(.) because of its
dependence on the real wage w/p and the real price of materials v/p.
Using the function Ft(.), we may rewrite the firm's optimization problem
at date s as:
(^11) max V5 - Js e (Mr)ptF(Kt) g (l.r)(K +6K )) dt
which yields the Euler equation familiar from the literatut-
I t (r + 6) l)(12) Ft'(Kt) - Pt( (l-r)
The right-hand side of (12) has traditionally been called the user Cost
of capital (e.g. Jorgenson 1963), for it defines the shadow price to whichi the
marginal product of capital should be set equal. However, wiLh other factors
of production the desired capital stock is a function of all input prices, not
just the direct input price of capital. Thus, if one is interested in
5
knowing the capital stock Itself, rather than its marginal product, an
alternative formulation of the user cost wili prove more uswfu:.
For purposes of exposition, let us assume that Ft0() ts the saparable
form:l
(13) Ft(K) - O(wt/pt, vt/pt)G(K) - StG(K)
Then, the first-order condition (12) may be rewritten:
g (r + s)(l-r)/(l-r)(14) G'(K t) - ct m =-: etpt
Because of the assumption that the firm can adjust its capital stock
instantaneously, expression (14) is a solution for the capital stock at date t
and, given the initial capital stock, the rate of investment as well.2
Therefore, since the function G(*) is time-invariant, the right-hand side of
(14) represents a sufficient statistic for the incentive to use capital in
production. We may think of this as the "full" user cost of capital. It
incorporates effects on investment working directly through the effective
rental price of capital as well as indirectly through the costs of other
factors of production.
B. The Effective Tax Rate
Many researchers (e.g. Auerbach 1983a, King and Fullerton 1984) have
found it useful to summarize the effects of the tax system on investment
through an "effective tax rate" calculation. Though most of the literature
has focused on developed countries, the approach has also been carefully
applied in the development context (Pellechio 1987). The thought experiment
6
giving rise to this measure Is to ask what rate cf tax applied to a broad.
based income measure would lead to the came wedge between after-tax and
before-tax returns as is actually observed. Put differently, for a given user
cost of cipital, what rate of tax on broad-based or "true economic" income
would lead to the observed after-iax return.
Despite its apparent simplicity, the concept does not give rise to a
unique definition, with the measure depending on which taxes are included in
the calculation and what level of after-tax or before-tax rate of return is
used as a benchmark. Moreover, the calculation of an effective tax rate alone
does not provide enough information to infer the effects of tax policy on
investment. Since the user cost of capital will result from adding the tax
wedge to the after-tax rate of return, it is important to know not only how
big the tax wedge is but to what extent it leads to a higher before-tax return
rather than a lower after-tax return. Even in small open economies that must
take world prices and rates of return as given, not all taxes will
necessarily be fully reflected in a higher cost of capital. Some will be
borne by imperfectly mobile factors (such as land and labor). Even with
perfect capital mobility some capital income taxes may be shifted abroad if
they are credited by foreign governments.
In spite of these limitations, the effective tax rate concept is a
popular one that can be useful for certain purposes, particularly in comparing
the relative incentive to invest in different assets. Therefore, we will
describe in somewhat more detail how it fits into the current framework.
One may think of the total tax wedge affecting the return to capital as
being divided into two parts. The first is the wedge between the required
rate of return r and the corporation's return before tax. The second is the
wedge between r and the return to investors after AUl taxes. The first wedge
7
is t'a effective rate of corporate tax, indicating how provisions directly
affectLng investment affect the corporate tax base. One may also think of
this as the effective rate of tax at the corporate level for an equity-
fLnanced Investment, ignoring any provisions permitting a deduction for
dividends paid.
To calculate the effective corporate tax rate, one would estimate how the
tax rate r in expression (14) would need to change to offset the repeal of
investment incentives and the imposition of a system of economic depreciation
allowances. This would involve varying r to offset the replacement of r with
the present value of economic depreciation deductions, r6/(r+6), holding al'
other terms in the expression fixed. The resulting effective tax rate
expression is (see Auerbach 1983a):
r(r+)(1-r)/(1-r) - 61 r(15) ec - (r+6)(1-r)/(i-r) - 6
where the denominator is the before-tax return to capital (equal to the
before-tax rate of return, net of actual depreciation) and the numerator is
the "tax wedge" between, this return and the corporate cost of funds r. This
expression provides the standard results that immediate write-off of assets
(r-r) leads to a zero effective tax rate and that with economic depreciation
allowances (for which r-6/(r+6)), ec - r.
A more comprehensive effective tax rate measure (see King and Fullerton
1984) is one that accounts not only for investment-oriented provisions at the
corporate level, but also the second wedge discussed above, between the rate
of return to firms after corporate taxes and the rate of return to savers.
This second wedge would account for interest deductibility at the corporate
level and taxes paid by individuals or other entities receiving the corporate-
8
source income. To get a total wedge equal to the sun of the two vedges, one
would add to the gap betveen the before-tax corporate return and the corporate
cost of funds r thle gap between r and the rats of return to suppliers of
funds, say s.
To calculate this total sffective tax rate, one must express r in terms
of the net return to savers. This is achieved by substituting expression (2)
into equation (14) and further expressing the interest rate I in terms of the
not real return to bondholders, say n, and the income tax rate on interest
received, say tp
The result is an expression for r in terms off the underlying real
raturns to equity and debt after all taxes, j and n, that can be substituted
into (15) to determine the total effective tax rata hat encoiapasses the tax
provisions embodied in r, the corporate tax rate t and the individual tax
rates 0 and tp Contrary to the previous case, one would measure the tax
wedge re'ative to p and n rather than r. Here the calculation depends on
which, of a variety of assumptions one makes concerning the ielationship of the
two net rates of return, p and n. The choice depends on which concept of
financial equilibrium one chooses (King and Fullerton 1984). For the "fixed
s" case in which these net returns are assumed to be equal (i.e., p - n -
s)3, this broader effectlve tax rate equals:
.r(r4-8)(l.r)/(l-t) . al -(17) OT - (r+6)(l.r)/(l.r) - 6
9
The numerator of (17) is the "total" tax wedge, incorporating the effects
of interest deductibility and personal taxes that manifest themselves through
the relationship of the corporation's cost of funds r and the net return to
asset owners, a.
The effective tax ra' OT describes the total tax burden on domestic
investm.nt, domestically financed. In a closed economy, it would therefore be
informative about the impact of zhe tax system on investment. In such an
economy, there is no distinction between taxes on saving and taxes on
investment. In a world with open economies, however, saving and investment
may occur in different places. Policies aimed at encouraging saving in a
country need not encourage investment there, but may simply cause more capital
to flow abroad. To the extent that the marginal investor supplies funds from
abroa-d, a different calculation that distinguishes taxes on saving and
investment may be necessary.
One approach would be to consicer the tax treatment of equity and debt
owned by foreigners, and include these in the calculation as well. For
example, Andersson et al (1990) calculate the effective tax rates eT for
investment in the United States financed not only by domestic debt and equity
funds, but also by debt and equity supplied via portfolio investment from
Japan. They likewise calculate the effeccive tax rates on Japanese
investment financed in Japan and from the United States.
The basic question to be addressed is how the firm's cost of funds r
relates to the required returns to equity and debt p and n when such funds
come from abroad. (For a small developing economy, the returns p and n may be
taken as fixed, so that the associated gap between the net returns p and n and
the gross return r translates directly into a higher cost of capital.)
10
The ansver to this question depends on both the host country's tax
treatment of such capital flows as vell as the home country's mechanism for
crediting forelgn taxes pald. This, in turn, depends on the type of entity
that is supplying the capital, for foreign direct investment by corporations
is treated differently than portfolio investment by individuals. Although
Andersson et_al treat the case of portfolio investment, foreign direct
investment and investment via financial intermediaries represent a more
significant portion of the flows between the United States and Japan. This
must be even more true of the capital flows into most developing countries.
It therefore seems most appropriate to consider the case in which the
investment is by a foreign corporation. We will discuss the implications of
this assumption further below.
C. The Effective Tax Rate and the User Cost of Ca2ital. More Broadly Defined
For developed economies such as Japan and the United States, the major
effects of policy on the incentive to invest may well come through the tax
system. Hence, the use of the various effective tax rate measures already
discussed may suffice. In developing countries, the most important effects of
policy may not work through the tax system at all, or may do so indirectly,
rather than through capital income taxes.
In terms of the user cost of capital expression on the right-hand side of
(14), we may distinguish between policies that affect the price of capital
goods, g, the required rate of return, r, the output price p and the
productivity term 8, through the effective real wage w/p or the effective cost
of material inputs v/p.
Policies affecting g and r may be seen as the equivalent of capital
income taxes, since they influence the gap between the gross and net returns
11
to capital. Put another way, they appear only in the first-order condition
for capital. (12), and not those for labor and materials, (6) and (7). For
purposes of measuring relative costs of capital and other inputs, one would
add only such policies to those previously considered, and the way of doing so
is straightforward. However, if one wishes to measure the incentive to
invest, then effects on p, w/p and v/p matter, too, since each of these
variables appears on the right-hand side of (14). For example, a subsidy to
labor or a protective tariff on an industry's output may well increase
investment. While it is misleading to equate such policies with a reduction
in capital income taxes, it is important to -onsider them along with policies
directed at capital specifically.
Some examples of how such policies affect the cost of capital dafined in
(14) follows.
1. Indirect taxes
If materials goods face on ad valorem tax rate tm, then the real
materials cost v appearing as an argument of e(.) (see (13)), would equal (1
+ tm)vw, if vw if the price net of tax (subscripted by w to indicate that this
will be the world price if other price distortions are absent). Assuming that
indirect taxes are not applied to exports, they will have no effect on the
expression for the output price p, which will equal the world price Pw
2. Tariffs
A tariff on materials inputs at rate T affects the cost of materials to
the firm just as an indirect tax would, v - (1 + T m)vw. However, a tariff at
rate Tp an output would raise the output price, relative to the world price,
to p - (1 + T p)pw. As is well known, this output price effect is equivalent
12
to a general production subsidy to the firm.
3. Dual Exchange Rates
If there is a controlled and an uncontrolled sector in the exchange
market, we may treat the difference between the two exchange rates as a
general trade intervention. Importers forced to buy foreign currency at the
(presumably higher) controlled rate are essentially facing a tariff.
4. Quantity Controls
In general, each type of quantity control has an analogous price
distortion. A well-known example is tariff and quotas. In this case, the
challenge is to identify the tariff-equivalent of the quota, which requires
some assumption about the price-elasticity of demand for the commodity in
question.
Other examples of quantity controls occur in the capital market. Here,
one can estimate the subsidy inherent in targeted funds by comparing the
stated interest rate to the market interest rate, as long as a latter such
rate is available. To the extent that such funds are used at the margin, the
implied subsidy rate should be used to adjust the interest rate appearing in r
(see '2)).
5. Imperfect Competition
If firms are not price-takers, this introduces the possibility of a mark-
up of the sales price p over marginal cost. The extent of the mark-up will,
of course, depend on the nature and degree of imperfection in the product
market.
13
One type of imperfection that Is relatively easy to analyze is
monopolistic competition, under which each firm faces a downward-sloping
demand curve with price elasticity i, where p depends both on the overall
elasticity of demand, the number of firms, and the degree to which import
substitution is possible. In thls case, the firm behaves as if it faces an
output price p (1- 1) rather than p in each of its factor utilization
decisions system. The case is analogous to that of production tax at rate v
In summary, policies affecting the numerator of the right-hand side of
expression (14) are capital-related; whether they are capital income taxes as
rypically included in effective tax rate measures or policies with similar
effects, they have equivalent marginal effects to a change in the rate of
capital income taxation. In this sense, they are appropriate for inclusion in
an accurate calculation of the 'effective tax rate" on capital income.
Tax and nontax policies that affect the denominator of the right-hand
side of (14) also affect investment and should therefore be considered in any
analysis that seeks to measure the full effects of policy on investment.
Though significant, their marginal effects differ from those of capital income
tax changes, for they also influence the real costs of labor and materials.
Moreover, because these polices affect the attractiveness of capital
indirectly through the price of output or other inputs, their impact on
investment cannot be measured without additional information about the
production process. That is, policies that affect p, w/p or v/p all work
through the term e in expression (14), and the form of e depends on the exact
specification of the production function, particularly the degree to which
other inputs are substitutes or complements for capital.
For example, suppose the production procE:s requires a fixed ratio of
materials to output and that value added by capital and labor is described by
14
a Cobb-Douglas function. Then X(.) has the form:
(18) X(K, L, M) - min(AKXLP, M/m)
for constants c, P and m, and 8(9) has the form (dropping time subscripts):4
(19) *(w/p,v/p)_tW 1/(1') l-m(v/p) 1/(1-0)
In this case, both labor and materials are complementary to capital in
the production process: an increase in either the real price of materials or
the real price of labor reduces the desired capital stock. The elasticity of
the user cost of capital, defined by the right-hand side of (14), with respect
to the real wage is 1/(l-p); the elasticity with respect to the real cost of
materials is m(v/p)/[l-m(v/p)). By comparison, the elasticity with respect to
the corporate tax rate r (holding r constant) is r/(l-r).5 (For more general
specifications of production, it will not even be possible to express F(s) in
the separable form given in (13) and the term e can only be locally
approximated).
Up to this point, all policies discussed have worked in markets with
fixed world prices. Policies driving a wedge between such world prices and
the prices facing the firm translate directly into changes in the user cost of
capital. One must add the marginal burden of capital income taxes to the net
returns required by suppliers of funds. Likewise, the domestic prices for
output and inputs, p and v, equal the world prices plus any tariff or tariff-
equivalent quantity restriction, such as an import quota, that is imposed
domestically. Unlike in a closed economy, no general equilibrium calculations
are necessary to estimate how much the gross return or price rises with the
15
tax. This makes the resulting effective tax rate more directly informative
about the user cost itself.
This simplicity is absent for labor market interventions, since (for most
countries) labor is not nearly as mobile as capital. Thus, one cannot
immediately compute the impact on the real wage rate and hence the user cost
of capital of tax and nontax policies that drive a wedge between the real wage
received by workers and the cost of labor facing firms. Incorporating the gap
between gros and nat wages in a grand "effective tax rate" computation may,
as a result, be extremely misleading if the incidence of labor income taxes
falls largely on workers rather than firms.
D. S
If one wishes to estimate the effects of tax policy on the incentive to
invest, the augmented user cost expression given in (14) provides a sufficient
statistic, given the modelling assumptions adopted in this section.
Traditional measures of the "effective tax rate" on capital fail in several
respects to provide an equally useful measure.
First, they typically ignore the separation of saving and investment
decisions and the importance of international capital flows. Second, they
consider only explicit taxes on capital and capital income, ignoring both
nontax capital policies (such as targeted lending) and tax and nontax
policies, such as tariffs and quotas, that indirectly influence the incentive
to invest through their effects on the prices of outputs and other inputs.
Finally, in emphasizing the magnitude of the tax wedge between gross and net
returns to capital, rather than the level of the gross return, a given
effective tax rate can correspond to several different levels of the desired
capital stock, depending on the incidence of the taxes in question. A given
16
tax wedge added to a price that is fixed in world markets may reduce
investment more than were the price determined domestically.
While the analysis to this point represents a useful summary of much of
the literature to date, it is static in nature. It therefore ignores the
dynamics of the investment process, a specification of which is necessary for
empirical work on investment. The characterization of the investment process
itself can be particularly important in cases where changes in the tax system
are being considered.
III. Changes in Tax Reglie
Over time, the economic conditions affecting investment may change quite
markedly. Among these economic changes are shifts in tax regime, caused not
only by policy shifts affecting all firms but also by shifts in an individual
firm's tax status. For example, a firm may face a zero marginal tax rate on
its taxable income for a period of years because it is carrving a large stock
of losses and depreciation allowances forward, and then become taxable once
again as these deductions expire or are used up. Both types of change in tax
regime, economy-wide and firm- or sector-specific, can exert a powerful, if
tsmporary impact on investment incentives. Indeed, in an unstable economic
environment, such "temporary" effects may nearly always be present. Thus,
one should go beyond examinations of tax systems applicable only in a "long
run' which is unlikely ever to occur.
Once one admits the importance of changes in economic conditions, the
assumption of instantaneous capital stock adjustment made above becomes even
more restrictive. It is clear that firms will not cause their capital stocks
to swinj wildly in response to each instantaneous change in the user cost of
capital. To model investment behavior realistically, therefore, it is
17
necessary to replace this assumptLon. The introduction of convex adjustment
costs for the capital stock provides such a smoothing incentive.6 'he
following analysis follows closely that first presented in Auerbach (1989).
For the interested reader, the full derivation is provided in the appendix.
An empirical application for the United States may be found in Auerbach and
Hines (1988).
We begin again with a firm seeking to maximize its value as in (11), but
introduce two changes, First, the tax parameters may vary over time, so that,
in particular, r, k and hence r require time subscripts. For the moment, we
continue to assume perfect certainty about these tax changes and the absence
of any risk. Second, we replace the exogenous price of capital goods g with a
total cost g(l - 60K + i ), chosen to give rise to a simple expression for
the marginal cost of capital goods:
1(20) q - d(g[l - 60K + W0(16K)]I)/dI - g(l + 0K)
The term the 0 is an adjustment cost parameter, equal to the percent increase
in effective capital goods prices to the firm per unit of additional
investment.
Replacing g in (11) with q as defined in (20), and adding subscripts to
the tax parameters, we obtain the following Euler equation for the firm,
replacing (12):
q (r+S)(l-r )/p - [q (l-r )/P(21) Ft (K )_t t t t t t
t t (1.r)
where the after-tax present value of investment incentives is:
18
aZ (r+*)(t-S)(22) r, - k + Js ta D (t-s) dt
Expression (21) is no more than a user cost of capital that ta'ces explicit
account of expected changes in the real, after-tax relative price of capital
goods q(l-r)/p (Auerbach 1983b). However, since q is a function of investment
itself, (21) is a first-order condition only rather than a direct solution for
K. To obtain the latter, one must substitute the expression for q given in
(20) into (21), obtaining a second-order differential equation in K that must
then be solved. Because this equation is nonlinear, a closed form solution
will not generally be available. However, such a closed form solution may be
derived if one linearizes the differential equation around its steady state
solution.7 The solution for investment may be expressed as a model of
partial adjustment toward a "desired" capital stock:8
.
(23) It - (- 1)(Kt Kt) + Kt
where the des' ed capital stock satisfies:
(24) G'(Kt) Ct -Jt2 e s2(s-t)c ds
the instantaneous cost of capital term ct equals
(25) ct - gt[(r + 6)(1-r ). (l-rt)
tp (l-rt)
and the terms el (s 0) and 02 (2 (r+6)) are the roots of the second-order
19
differential *quation.9 (As before, the function C(e) is defined in
expression (13) as the production function divided by the term B.)
Because the weights o2a 2(St) sum to one, we can viow expression (24)2
as indicating that the desired capital stock that influences investment at
date t depends on a weighted average, Ct, of present and future user costs of
capital. Only if adjustment costs are zero and hence adjustment is
instantaneous (in which case a2 - a) or if the cos. of capital is constant
over time will the current cost of capital be sufficient to describe the
effects of the tax system on lnvestment. In general, forward looking
investment behavior that depends on the weighted average of current and future
costs of capital may be quite different from that implied by assuming a
constant cost of capital without changing tax rates. The use of this new
methodology is straightforward. It differs from traditlonal specifications
primarily in its dependence on predicted future capital costs rather than
lagged ones. To apply it, one first calculates the instantaneous user cost of
capital at each date t, ct, and then aggregates these user costs over all
future dates. The weights to use in this aggregation depend on a number of
parameters (see footrote 9), not all of which are precisely known (such as
0). Hence experimentation with different weighting scheme seems called for.
In the firsz step, one must allow for potential changcs in the tax rate r when
calculating r, and must also allow for potential changes in r itself.
We now provide some examples to illustrate this approach. It is helpful
in making these examples realistic to draw them from the policies and
experiences of particular countries. However, such examples should not be
interpreted as an overall evaluation of the tax policies of the countries in
question.
20
Many countries have recently enacted tax reforms aimed at broadening the
tax base whlle at the same time loworing tax rates. The effects on Investment
of the 1986 U.S. reform are dlscussed in Auarbach (1989).
Among developing countries, Mexico has recently moved to an indexed
corporate tax system, with a phased reduction in the corporate tax rate from
42% to 35%. During the transition period, the tax rate reduction itself has
three effects on the instantaneous user cost of capital given in expression
(25). First, it reduces the tax rate term appearing in the denominator,
lowering the cost of capital. Second it reduces the after-tax present value
of depreciation deductions, r (calculated using (22)), increasing the cost of
capital. Third, it makes r, the time derivative of r, negative: the present
value of depreciation deductions declines over the period as the tax cut is
phased in. This last effect reduces the user cost: there is an incentive to
invest while depreciation allowances may still be deducted at the higher tax
rate. on balance, the instantaneous user cost, as well as the weighted
average of current and future such user costs relevant to current investment,
will likely fall, stimulating investment.10 It is even possible that
investment will be stimulated more by a phased reduction in the tax rate
rather than an immediate one, since the anticipated decline in the value of r,
by itself, stimulates investment. 1 1 This possibility emphasizes the
distinction between average and marginal tax rates, between the level of taxes
paid by a company and its incentive to invest. A delayed reduction in the tax
rate r will certainly cause the firm to pay more taxes in the short run, even
in it faces a lower cost of capital and hence invests more.
A similar distinction may be made between the effects of investment-
oriented incentives, such as investment tax credits and allowances, and cuts
21
in the tax rate r. While both will spur investment, rate reductions will
reduce tax payments by more, given the level of investment stliulus, because
they will also reduce the taxes the firm pays from sources of income other
than new investment, including existing capital and economic rents.
B. Tax holid v
Many countries provide tax holidays to attract new investment. Tax
holidays provide the investing firm with an exemption from tax on its
normally taxable income during some time period after the firm's initial
investment is made. As discussed in Mintz (1989), such a holiday does not
necessarily imply that the firm's user cost of capital is the same as it would
be in the absence of taxation, since the holiday is not permanent. In
considering whether to invest, the firm must calculate the taxes it will pay
on today's asset purchase once the holiday is over, as well as the tax
incentives to invest at a later date. Neither of those factors would apply if
the holiday were permanent, for all present and prospective investments.
The problem of tax holidays can be analyzed in exactly the same manner as
the "global" tax rate change just considered. The situation is the same as if
the firm faced a zero tax rate for a predetermined length of time, followed by
the normal rate of tax r, thereafter.
Once again, there are three effects on the instantaneous user cost of
capital during the holiday period. The tax rate at the current date is zero.
To the extent that the depreciation allowances on the firm's current
investment extend beyond the holiday period, the present value of after-tax
depreciation allowances r would be reduced but not eliminated.1 2 Finally, the
time derivative of this present value, r, would be positive, since the
fraction of allowances deductible from tax would rise as the end of the
22
holiday period approached. The first impact would be positLve, the second
and third negative. The impact on investment during the holiuay period would
depend on the generosity of the Investment incentives themselves. It is
entirely possible that some types of investment would be discouraged. This
woold be most likely in cases where the initial investment allowances were
larger than concurrent cash flow. In such cases, new investments would
generate a negative tax base in the years immtdiately after an investment, so
firms would actually benefit (with respect to their nDw investment) by being
taxable. See Auerbach (1983a).
The revenue cost of a tax holiday depends on whether it applies to assets
already in place. If it does then, like a permanent tax rate reduction, the
tax holiday reduces the taxes that firms pay during the holiday period on
preexisting sources of income other than the new investment that the policies
aim to encourage. This makes the tax holiday more costly than more targeted
investment incentives, such as investment tax credits or grants.
C. Tax law asyuetries
Most countries alloo firms with net operating losses to carry these
losses forward, to be used to offset subsequent taxable income. Some
countries also allow losses to be carried back, providing refunds against
taxes previously paid. Firms that are currently not paying taxes but with
some probability will be doing so in the future can be treated as if they are
facing a tax regime with marginal tax rates that change over time. In this
sense, the case is similar to the previous one of tax holidays. However, in
this case, one cannot simply assume a current marginal tax rate of zero for a
firm that is not presently paying taxes. In present value, additional income
earned today may well lead to a significatit tax liability, even if no taxes
23
are paid Immediately.
For example, suppose a firw has a tax loss this year, which it will carry
forward and, with certainty, use up next year, when it will be paying taxes
once again. If the firm generates another dollar of income this year, this
income will reduce the tax loss carried forward by one dollar. This
reduction, in turn, will increase by one dollar the firm's taxable income the
following year, since the size of the deductible tax loss will be smaller.
Hence, the firm will pay taxes on an additional dollar of income with a one
year delay. The true marginal tax rate for the current period, which one may
think of as " "shalowl tax rate, is therefore the statutory rate, r,
discounted for one period at the nominal interest rate.
OJf course, one cannot be certain of the date at which a firm not
currently paying taxes will begin doing so, but this does not pose a
conceptual problem for the application of the preceding methodology. If one
can estimate a probability distribution of when a firm will begin paying taxes
again, one can simply multiply the tax rate for each date by the associated
probability and add the discounted values of these products together to obtain
today's shadow tax rate. Doing so for each date, one can produce a time
profile of shadow tax rates for a given firm, which can then be used to
calculate the user cost of capital.
Illustrations of this approach are presented in Auerbach (1983a),
Auerbach and Poterba (1987) and Altshuler and Auerbach (1989). It can be
applied even in cases where firms are permitted to carry losses back, and
where different rules for carrying forward apply to different components of
taxable income. In the United States, for example, the rules for carrying
forward unused investment tax credits have differed from those applying to
ordinary losses. In other countries, such as Pakistan, net operating losses
24
exclusive of depreciation allowances can be carried forward for only six
years, while unused depreciation allowances themselves can be carried forward
indefinitely. Hence, the shadow tax rate applicable to depreciation
deductions should be closer than the shadow tax rate applicable to ordinary
income to the statutory tax rate. In Mexico, the value of losses carried
forward is indexed fov inflation. Therefore, the deferral of tax payments
should be discounted by a real rather than a nominal interest rate when
computing shadow tax rates.
The importance of allowing for tax losses and related asymmetries depends
on the empirical significance of such phenomena. In the United States, for
example, Altshuler and Auerbach (1989) estimated that firms faced an average
marginal shadow tax rate of 32% in the early 1980s even though the statutory
marginal tax ratc during the period was 460. The importance of tax losses
has been demonstrated for Canada as well (Mintz 1988).
As with tax holidays, a temporary respite from taxes induced by tax loss
carryforwards can have complicated effects on the incentive to invest. If a
program of generous investment incentives is in place, investment by firms
that are not paying taxes may actually be discouraged. In such situations,
alternative forms of investment incentives may be desired, such as direct
grants that do not work through the tax system.
D. Uncertainty and risk
As the discussion of tax law asymmetries illustrates, uncertainty about
the tax regime a firm will face in the future may exert a significant impact
on its current investment. A realistic treatment of the effects of tax policy
must also account for the uncertainty that firms will attach to government
policy itself. Countries without an established reputation for following
25
through on announced policies may face difficulties making Investment
incentives effoctive. The possibility of dynamic inconsistency on the part of
governments has played a role in past discussions of the design of tax policy,
suggesting why generous tax holidays might be necessary to attract foreign
investment (Doyle and van Wijnbergen 1984).
This issue has several implications for the cost of capital
specification. First, anticipated tax rates that appear in the cost of
capital expression should not necessarily be those listed in government
documents. One must allow past behavior to inform the determination of such
tax rates. Second, the efficacy of a tax policy should be judged with respect
not to its announced changes, but rather the changes it induces in the policy
anticipations of investors. One policy may appear more stimulative than
another, but may be found to be less plausible or permanent. For example, a
promised reduction in the tax rate * may be reversed or postponed more easily
than an investment tax credit already given today can be taken back from the
taxpayer in the future (Hansson and Stuart 1989). Finally, the uncertainty
with respect to tax policy may cause risk-averse investors to demand a risk
premium, a higher rate of expected return. Hence, a climate of uncertainty
about tax policy may, in itself, discourage investment. More generally, risk
is a central aspect of the investment process. Even with a riskless tax
environment, investors may be subject to considerable uncertainty about the
future profitability of their prospective investments, and may as a result
demand an expected rate of return considerably in excess of the risk-free
interest rate. The required rate of return r that appears in the user cost of
capital expression (25) derived above must reflect this risk premium.
Likewise, the rate of discount applied to future depreciation allowances must
account for any risk associated with such tax benefits. Indeed, there is
26
nothing requiring that the discount rates appropriate for tax benefits and
other after-tax flows be the same. While such differences may be easily
accommodated in the cost of capital calculation, they make standard effective
tax rate calculations based on a uniform rate of return inappropriate and
potentially quite misleading (Auerbach 1983a). The discount rates applicable
to future tax benefits are especially important in the design of tax
incentives.
IV. IhQUAhXLCtt:&lCulating r and,
To implement the model of invescment behavior derived in the previous
section, one must incorporate the relevant tax and nontax provisions affecting
the firm's required rate of return r and the present value of its investment
incentives, r.
A. Measuring investment incentives
Most countries provide schedules of straight-line or declining balance
depreciation allowances. Such schedules may be extremely accelerated relative
to actual economic depreciation. Turkey, for example, provides a 50%
declining balance depreciation rate for equipment. However, these allowances
are typically not indexed for inflation, and so must be discounted using a
nominal ldscount rate. Mexico has recently introduced the choice of a one-
time, first-year deduction, in lieu of all subsequent depreciation allowances,
that is meant to provide roughly the present value of such depreciation
allowances and protect them from inflation.13
In addition, many developing countries provide initial relief for
investors over and above normal depreciation deductions. In Turkey, for
example, there are investment allowances of 30% to 60% for certain types of
27
investment. In Pakistan, the initial allowance for machlnery and equipment is
40%, with the allowance deducted from the basis used for subsequent
depreciation.
Other investment incentLves do not fit as dlrectly into the expression
for r given above, but may be expressed in equivalent terms. For example, the
value of a subsidized loan assocLated with a particular investment may be
computed by estimating the present value after-tax of the interest and
principal payments made on the loan and subtracting this from the face value
of the obligation, i.e. the amount of money initially provided to the
lnvestor,
A soraewhat more complicated investment scheme ls the 'investment fund" or
(as it is referred to in Turkey) "financing fund' system. Such a scheme
provides firms with a tax deduction for setting funds aside in the investment
fund. The funds may subsequently be drawn down for the purpose of making
investments. Thelr use in Sweden has been the subject of previous discussion
in the literature (e.g. Taylor 1982, Sodersten 1988).
In Turkey, firms may contribute up to 25% of their taxable profits to the
fund in a given year and receive a deduction for doing so. The funds are
deposited in a government bond account at the central bank, and may be drawn
down to the extent of new investment in the future. However, the firm must
add the contributed funds back into taxable income one year hence, so that it
receives a one-year tax deferral on the contribution regardless of how long
the funds remain in the account. Balancing the benefit associated with this
tax deferral may be the cost of keeping funds in an account yielding what may
be a below-market interest rate. Even if a net tax benefit remains, a serious
question remains abouz the efficacy of such a program in stimulating
investment.
28
The problem with investment fund schemes is that the tax benefit may well
be unrelated to the aaxguIl investment decision. If firms are investing at
least a quarter of their earnings anyway (this is not an especially high rate
of reinvestment), then the scheme in practice is equivalent to one that simply
gives firms a one year tax deferral on a quarter of their earnings in exchange
for placing these earnings in a government account for a year. While this
scheme may benefit the firm, it does not provide any subsidy to new
investment. It encourages investment only in the sense that it reduces the
effective tax burden on 25% of the fuiture earnings that such investment
generates, in precisely the way that a very small reduction in the rate of
income tax r would.
B. heasuring the recuired rate of return
There are several issues relating to the measurement of the required rate
of return r. Already discussed above is the need to use realistic rates of
return that reflect the risk premia required by the market. In an economy
with well-developed financial markets and most investment undertaken by public
corporations, the required nominal return to debt i in expression (2) would
be well approximated by the observed nominal interest rate, and the required
return to equity before personal taxes, /(l-H), could be based on observed
returns to equity. One could use either an after corporate tax earnings-price
ratio or a market return (dividend yield plus capital gain) for this purpose.
A benefit of this approach is that it may not require one to specify the tax
treatment of those who supply the funds to corporations.14
In a developing country, such returns to debt and equity may not be as
easily observable. In this case, one may need to use information on world
interest rates, combined with the tax rules that apply to foreign source
29
capital Income. For Oxample, suppose the U.S. interest rate ls i* An
AmoerLcan investing in foreign debt must pay whatever taxes are withheld abroad
on the repatriated Lnterest income, plus U.S. taxes net of any allowable
foreign tax credit. (If the foreign taxes are fully creditable, then the U.S.
investor bears only his U.S. tax rate on the interest received.) Let tp be
this U.S. tax rate. Then the investor's net return in the U.S. will be
i*(ltp) - w*, where w* is the U.S. inflation rate. Assuming that exchange
rate gains and losses are not taxable, the dollar rate of return available
abroad will be i(l-tmax) - w - d, where i is the foreign nominal interest
rate, tmx is the higher of tp and the rate of withholding tax, a is the
foreign inflation rate and d is the rate of foreign currency depreciation
against the dollar.
Equating these two net rates of return yields:
(26) i i*(l-tp) - ( a* - f-d)(1-t Max)
In cases where the liability is denominated in dollars, the term
(i* - - d) vanishes because all transactions are in the same currency. (The
term will also vanish if purchasing power parity is satisfied.) If, in
addition, taxes withheld are fully creditable, then t - t and i - i*max p
More generally, however, both of these sources of difference between i and i*
will be present. The low rates of income tax now in effect in many developed
countries (including the United States), may in some cases be exceeded by
foreign rates of withholding tax on interest. For example, while the top tax
rates in the United States are 34% for corporations and 33% for individuals,
Mexico withholds 42% of interest payments. Further, some countries follow the
30
territorial approach to taxation and offer no credit at all for foreign taxes
withheld. Even after one allows for the effects of these tax provisions, it
is still necessary to account for differences in risk among countries that
would be reflected in required rates of return after-tax. However uncertain
one is about the size of such risk premia, expression (26) is still useful
because it shows how changes in the domestic withholding tax rate affect the
cost of capital, given the level of risk.
Computing the required return to equity by using rates of return observed
abroad is even more problematic than in tha case of debt. First of all, since
equity normally bears a considerably greater fraction of investment risk than
debt, the problem of measuring risk premia is more significant in this case.
Second, the tax treatment of equity investment from abroad is more
complicated than the treatment of debt. The effective rate of tax depends not
only on the rates at which taxes are withheld and credited, but also on
whether the funds come from another corporation via foreign direct investment
or from the household or banking sector, and whether the equity funds for
investment abroad come from earnings retained from existing projects abroad or
new equity contributions from the investing country (see Hartman 1985, Gordon
1986). A comprehensive discussion of this problem is beyond the scope of this
paper. However, one may cite some basic results that are helpful in guiding
the specification of the required return to equity.
Consider the case of foreign direct investment, in the "host" country.
Let v - p/(1-0) be the required return to equity in the country from which the
funds come, the "home' country, and ignore for the moment differences in risk.
If such funds are sent abroad and all their earnings repatriated, the rate of
return after taxes in the host country must equal v plus any additional taxes
imposed in the home country upon repatriation, net of foreign tax credits. If
31
tf is the foreign tax rate imputed by the home country for such receipts from
abroad, and t is the home country's corporate tax rate on repatriated
earnings, then the required return to equity abroad after foreign taxes will
be v(l-t in)/(l-tc), where tm* is the smaller of tf and tc If the foreign
tax rate used when imputing the credit (typically not the statutory tax rate r
but some estimate of the presumably lower effective corporate tax rate in the
host country) is at least as high as the home country's tax rate on foreign
earnings, t*, then this required rate is just the rate of return required atc
home, v: no further corporate taxes will be owed in the home country. If
additional taxes are due on repatriated earnings, however (this will never be
the case for home countries following the territorial approach, where tc - 0)
then the required return to equity will exceed v.
When foreign direct investment is funded by retained earnings already in
the host country, however, the calculation of the required return to equity v
is simpler. In this case, the tax treatment of repatriated funds is
irrelevant, since repatriated funds will bear the same rate of tax and, in
present value, the same tax burden regardless of when they are repatriated
(liartman 1985). Thus, the required rate of return will always be v.
Therefore, for countries with tax rates sufficiently high to provide enough
foreign tax credits to offset further corporate tax liability upon
repatriation, the required rate of return to equity after corporate taxes
(except for differences in risk) will be the required rate of return to equity
in the countries supplying the investment funds.
Thus, for both debt and equity, the major difficulty involved in
estimating the firm's required return is the estimation of the domestic risk
premium.
32
V.
This paper has reviewed the literature on investment and the cost of
capital, showing how the effects of tax and nontax government policies should
be incorporated in the analysis of investment behavior. The methodology is in
several respects more general than calculations of tax wedges and effective
tax rates. Its application in a developing-country context should provide
light on the ability of policy to influence investment, the efficacy of those
policies currently being pursued, and the appropriate directions for reform.
33
Table I
Notation
X(.) - general production function, with capital, labor and materials asarguments.
F(-) - residual production function, with capital as an argument, obtainedby subtracting labor and materials costs from X(n) and solving forlabor and materials as function of K. (defined in equation (10)).
G(.) - residual production function normalized for fluctuations in theprofitability of capital (defined in (13)).
e - term representing the fluctuation in the profitability ofcapital due to variation in input prices. (defined in (13)).
6 capital depreciation rate (geometric).
i - nominal interest rate.
b - debt-value ratio.
x - inflation rate.
I' - required real after-tax return to equity holders.
v - -p/l-O real required return to equity, before personal taxes.
n . real return to bondholders after tax (defined in (16)).
r . weighted average cost of capital (defined in equation (2)).
p - output goods price.
pW - world output goods price.
v - material goods price.
vm - world material goods price.
g - capital goods price.
q - capital goods price, including marginal adjustment costs.
0 - adjustment cost parameter.
w - nominal wage rate.
9 - effective household tax rate on equity income.
k - investment tax credit.
34
D(a) * depreciation deduction for a capital good of age a.
r present value of investment credits and dopreciation deductions(defined in equation (3)).
*c effective corporate tax rate (defined in (15)).
aT ' *effective total tax rate (defined in (17)).
tp - personal tax rate of bondholders.
Tm tariff rate on material good.
Tp tariff rate on output good.
tX excise tax on material good.
35
This appendix sketches how the decision rule given in equations (24) and
(25) can be derived as a solution of the linearized version of the Eular
equation (21).
For simplicity, we normalize the output price, p, to one. First (letting
Ft(K) - BtG(K)), express (21) as a differential equation in 4:
(Al) 4 - - G'(K)G("r.) + q(r+5) -q(l. r)1-r 1-r
Linearizing around the steady state (where q - g and r - o), we obtain
(letting * denote steady state values):
(A2) 4 u -G"(K ) (1-r ) (K-K ) + (r+S)(q-g)
l.r*
- KD * ((l-v)-(l-v )j , - K)1 (B- )G(K- ) (1-F
+ G* [(1 ) (1-F r) g *l-(1-r ) (-
Using expression (20) for q and the fact that G'(K ) - c - g (r+. ) (l-r ),
we obtain:
(A3) Kt '(r+6) K t ~t00
where a - -G"/G', xt _ °(r+ KJ 11 - a*t, and
36
(A) at (1-,). (1v ) + (-rt) ' (1-r) 0 0 * 1 (1:r)
(1- t) (1- r) 6* r+5 (l-r )
The equation (A3) has roots:
(A5) r+6 + .(r+6)2 + 4a(r+g)- 0 i- 1,2
2
Solving the unstable root, °2> 0, forward, one obtains the first-order
equation:
2 (8-t)(A6) Kt °1 K t - 2 (t-t da
which may be rewritten as the partial adjustment model given in expressi-
A * nt(23) in the text where Kt- K a and
_ ZU-o2(s-t)(A7) a - r ° * a ds
t t 2
By another first-order approximation,
A * A
(A8) G (K)- G'(K) + C(K) (Kt -K)
* A *
- G'(K )(1 - °(Kt - K ))
_ G'(K )(l+0 t)
37
Substitution of (W7) and the value of GI (K ) into (AS) yields:
A *u2 (s-t) *(A9) G(K t) J e 22 a lg(r+8)(l-rL) (1 + as)) ds
However, from inspection of (A4), we observe that a is simply the
flrst-order deviation of c5 as defined in (25) from c
*(AlO) cs - &(r+8)(l-r ) (1 + a)
S (l-r )
Substitution of (A10) into (A9) yields expression (24) in the text.
When there are constant returns to scale in production, a1 - - 0.
Hence, the solution based on (A6) is:
(All) Kt - r 2 (st aa ds - 2 e s2 a ds0 t 0
-a (s-t)co 2 * 1
-I I o e c (l+a) ds + -
c*0 t 2 0
-o (s-t)
1 1 rae 2 c ds- J~t 2 s d
0 c*
Once again, investment depends on current a-d future values of the
instanteneous user cost of capital, c.
38
R9 mu
Altshuler, Rosanne, and Alan J. Auerbach, 1989, 'The Importance of Tax LawAsymmetries: An Empirical Investigation," Ouarterly Journal of Economics,forthcoming.
Andersson, Krister, Kenji Aramaki, A. Lans Bovenberg and Sheetal K. Chand,1990, "Tax Incentives and Irnternational Capital Flows: The Case of the UnitedStates and Japan," in Assaf Razin and Joel Slemrod, eds., InternationalAsLects of Taxation, Chicago: University of Chicago Press, forthwoming.
Auerbach, Alan J., 1983a, "Corporate Taxation in the United States," BrookinusPaRers on Economic Activity, 2, pp. 451-505.
----------, 1983b, "Taxation, Corporate Financial Policy and the Cost ofCapital," Journal of Economic Literature, September, pp. 905-940.
----------, 198v, 'Tax Reform and Adjustment Costs: The Impact on Investmentand Market Value," International Economic Review, December, forthcoming.
.-------- and Dale W. Jorgenson, 1980, "Inflation-Proof Depreciation ofAssets," Harvard Business Review, September/October, pp. 113-18.
---------- and James R. Hines, 1988, "Investment Tax Incentives and FrequentTax Reforms," American Economic Review, May, pp. 211-16.
--------- and James M. Poterba, 1987, "Tax Loss Carryforwards and CorporateTax Incentives," in M. Feldstein, ed., The Effects of Taxation on CapitalAccumulatin, pp. 305-38.
Doyle, Chris and Sweder van Wijnbergen, 1984, "Taxation of ForeignMu?'inationals: a Sequential Bargaining Approach to Tax Holidays," InstitutefcL International Economic Studies seminar paper no. 284.
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Hartman, David, 1985, "Tax Policy and Foreign Direct Investment," Journal ofPublic Economics, February, pp. 107-21.
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40
Footnotes
1. A particular example of a production structure giving rise to F(s) havingthis separable form is given below.
2. As has long been recognized, a problem arises if the underlyingproduction function X(.) satisfies constant returns to scale. In that case,the derivative of the production function F(s) is not a function of K, so thatexpression (14) is overdetermined. In this case, the optimal capital stock iseither zero, infinite or indeterminate, depending on whether or not (14) issatisfied. It is therefore necessary to assume either decreasing returns toscale in K, L and K, that the capital stock cannot be adjustedinstantaneously, or that the firm's marginal revenue curve is not horizontal.The latter two assumptions also make the characterization of the firm'sdecision more realistic. This is discused further below.
3. One could also assume equal before-tax returns (the "fixed-p" case) orequal intermediate returns (the "fixed-r" case). The problem of choosingamong these assumptions is due to the fact that debt and equity coexist eventhough the tax wedges imposed on debt and equity returns differ. Thishighlights a limitation of the procedure, its ignorance of risk and otherconsiderations that might help explain observed patterns of financialstructure and asset ownership.
4. The whole function F(-) has the form G(.)G(w/p,v/p), where
G(K) _
5. Given this formulation, one can readily see the relationship of thisdiscussion to the concept of effective protection. Given fixed world pricesp for the output good and vw for the input good, the institution of tariffsTp on the output good and Tv on the input good causes the term pg in thedenominator of (14) to equal
pW (1+T ) w'1l/(# )[1-m(vw/pw)(l+Tv )/(l+Tp)
If we define T to be the uniform tariff that provides the sameprotection for the Industry and hence the same desired capital stock, weobtain:
[1 - m(v/p) 1(l+T )/(l+Tp)]] 1/(1 )To - (1 + T ) -e P ~~~1 m (v/p)]
which is less that T if and only if T > Tp. The relationship of effectiveprotection to effectYve tax rates is dYrcussed by Guisinger (1988).
41
6. Although the result will not be used here, the convex adjustment costmodel can also be used to provide a rigorous underpinning for the "q" theoryof investment first envisioned by Tobin (1969), under which the firm'sinvestment behavior is related to its market value (Hayashi 1982). Given themarket value of the firm, one can then regress investment on the tax-adjusted q ratio of market value to asset replacement cost to obtain estimatesof the adjustment cost function (Summers 1981). Unfortunately, this approachdoes not permit one to measure directly the impact of future costs of capitalon investment.
7. An alternative approach, found in Pindyck and Rotenberg (1983), is toestimate the production function and adjustment cost parameters directly fromthe Euler equation, without solving for the underlying investment rule. Thatis, instead of solving for an expression for K that is not a function of R,they estimate the Euler equation obtained by substituting (20) into (21) withinstrumental variables, treating K as an endogenous regressor. Like theapproach of estimating the Euler equation based on (21) alone, i.e..regressing investment on q, this technique does not provide any insight intothe effects of future costs of capital on investment.
A
8. When there are constant returns to scale in production, K is eitherinfinite, zero or indeterminate. However, even in the former two cases, therate of investment will still depend on the costs of capital as defined in(24).
9. Given the formulation of the problem, these roots have the form:
ai - (r+6) .(r+6)2 + 4e(r+6)/0 i - 1, 22
where e - G"/G' at the point of linearization. When there are constantreturns to scale, e - 0. If, however, the firm faces a downward-slopingdemand curve, then the relevant elasticity e would be based on pG rather thanG. The negative relationship between price and output would impart morecurvature to the marginal revenue product dSpG). Even with G" - 0, there would
dKstill be curvature in the revenue resulting from additions to the capitalstock.
10. A full analysis of the Mexican reform would be considerably morecomplicated, for it would require inclusion of the program's other changes,notably the effects indexation of depreciation allowances and interestpayments. The former effects would be incorporated via the allowances D(e) inthe calculation of r and r using (22), and the latter would be treated throughinduced changes in the corporate cost of funds r.
11. For further discussion, see Auerbach (1989).
12. This assumes that the firm cannot defer depreciation allowances occuringduring the holiday period. If they can, there would be a much smaller declinein r, due only to the discounting of these delayed deductions. In such a
42
case, the Incentive to in-fost during the holiday period would be much greateran r would be larger and r smaller. See Mintz (1989) for further discussion.
13. Such a scheme, and its advantages, is discussed in Auerbach and Jorgenson1980.
14. This simplicity is based on the q - 10 assumption thaF a dollar investedby the firm costs shareholders a dollar and that a dollar of earnings is wortha dollar to shareholders whether distributed or not. Under the "taxcapitalization hypothesis, the ratio of shareholders' value to firm value, q,may be less that one, equal to the ratio of the after-tax proceeds of a dollardistributed to those of a dollar retained by the firm. In this case, anappropiate measure of equity cost based on observed earnings would multiplythese earnings by q. (see Auerbach 1983b), to offset the multiplication by qalready implicit in the firm's value. To make this correction, however, onewould have to know the tax rates of the 'representative" shareholder.
Whether the "q - 1" or 'tax capitalization" view depends on the firm'smarginal source of funds. If the firm finances its marginal investments usingretained earnings, it faces a lower cost of capital because a dollar of fundsretained does not cost taxable investors one dollar. There may be otherreasons why firms face a lower cost of capital when using internal funds, forexample becau e of information asymmetries. One way if identifying whichequity regime a firm is in is by the level of its internal funds. Within thecost of capital framework, one could posit that some function of cash flowdetermines the appropiate adjustment to observed earnings-price ratios (i.e.,whether to multiply earnings by some value of q < 1.) An alternative, more adhgl approach, has been to put cash flow separately into the investmentequation. Doing so has recently been found to be quite significant inexplaining the investment behavior of smaller U.S. firms (Fazzari, Hubbard andPetersen 1988).
43
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